scholarly journals Construction of exact minimal parking garages: nonlinear helical motifs in optimally packed lamellar structures

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200891
Author(s):  
Luiz C. B. da Silva ◽  
Efi Efrati
Keyword(s):  
Minimal Surfaces ◽  
Arbitrary Distribution ◽  
Analytical Treatment ◽  
Lamellar Structures ◽  
Left Handed ◽  
Area Functional ◽  
Parking Garages ◽  
Lateral Displacements ◽  
Helicoidal Structure ◽  
The Second Variation

Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are comprised of such minimal surfaces in which right- and left-handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter-motif separation is comparable with its pitch, and thus this approximation fails. Here, we present a recipe for constructing exact minimal surfaces with an arbitrary distribution of helical motifs, showing that any harmonic graph can be deformed into a minimal surface by exploiting lateral displacements only. We analyse in detail pairs of motifs of the similar and of opposite handedness and also an infinite chain of identical motifs with similar or alternating handedness. Last, we study the second variation of the area functional for collections of helical motifs with asymptotic helicoidal structure and show that in this subclass of minimal surfaces stability requires that the collection of motifs is pitch balanced.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

scholarly journals Cell wall composition determines handedness reversal in helicoidal cellulose architectures of Pollia condensata fruits

2021 ◽  
Vol 118 (51) ◽  
pp. e2111723118
Author(s):  
Yin Chang ◽  
Rox Middleton ◽  
Yu Ogawa ◽  
Tom Gregory ◽  
Lisa M. Steiner ◽  
...  
Keyword(s):  
Cell Wall ◽  
Cell Walls ◽  
Hierarchical Structures ◽  
Cell Wall Composition ◽  
Cellulose Microfibrils ◽  
Plant Cell Walls ◽  
Mechanical Responses ◽  
Structural Colors ◽  
Left Handed ◽  
Helicoidal Structure

Chiral asymmetry is important in a wide variety of disciplines and occurs across length scales. While several natural chiral biomolecules exist only with single handedness, they can produce complex hierarchical structures with opposite chiralities. Understanding how the handedness is transferred from molecular to the macroscopic scales is far from trivial. An intriguing example is the transfer of the handedness of helicoidal organizations of cellulose microfibrils in plant cell walls. These cellulose helicoids produce structural colors if their dimension is comparable to the wavelength of visible light. All previously reported examples of a helicoidal structure in plants are left-handed except, remarkably, in the Pollia condensata fruit; both left- and right-handed helicoidal cell walls are found in neighboring cells of the same tissue. By simultaneously studying optical and mechanical responses of cells with different handednesses, we propose that the chirality of helicoids results from differences in cell wall composition. In detail, here we showed statistical substantiation of three different observations: 1) light reflected from right-handed cells is red shifted compared to light reflected from left-handed cells, 2) right-handed cells occur more rarely than left-handed ones, and 3) right-handed cells are located mainly in regions corresponding to interlocular divisions. Finally, 4) right-handed cells have an average lower elastic modulus compared to left-handed cells of the same color. Our findings, combined with mechanical simulation, suggest that the different chiralities of helicoids in the cell wall may result from different chemical composition, which strengthens previous hypotheses that hemicellulose might mediate the rotations of cellulose microfibrils.

scholarly journals On minimal surfaces on two-step Carnot groups

2019 ◽  
Vol 485 (4) ◽  
pp. 410-414
Author(s):  
M. B. Karmanova
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Carnot Groups ◽  
Correct Formulation ◽  
Area Functional ◽  
Suitable Notion ◽  
Minimality Conditions

For graph mappings constructed from contact mappings of arbitrary two-step Carnot groups, conditions for the correct formulation of minimal surfaces’ problem are found. A suitable notion of the (sub-Riemannian) area functional increment is introduced, differentiability of this functional is proved, and necessary minimality conditions are deduced. They are also expressed in terms of sub-Riemaninan mean curvature.  

scholarly journals On the existence of min-max minimal surface of genus g ≥ 2

2017 ◽  
Vol 19 (04) ◽  
pp. 1750041 ◽  
Author(s):  
Xin Zhou
Keyword(s):  
Variational Method ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Minimal Surfaces ◽  
Strong Convergence Theorem ◽  
Plateau Problem ◽  
Mountain Pass Lemma ◽  
Area Functional ◽  
Genus Formula ◽  
Direct Variational Method

In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus [Formula: see text]. We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321] and Rado [On Plateau’s problem, Ann. Math. 31 (1930) 457–469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-[Formula: see text] minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding–Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding–Minicozzi and the author to all genera.

Unsteady flow near a moving cylinder

1994 ◽  
Vol 278 ◽  
pp. 363-390
Author(s):  
A. P. Dowling
Keyword(s):  
Boundary Layer ◽  
Lateral Displacement ◽  
Pressure Fluctuations ◽  
Unsteady Motion ◽  
Arbitrary Distribution ◽  
Radius Of Curvature ◽  
Cylinder Axis ◽  
Flexible Cylinder ◽  
Moving Cylinder ◽  
Lateral Displacements

General representations are derived for both the velocity potential and the surface pressure fluctuations induced by an arbitrary distribution of vorticity near a manoeuvring cylinder. The cylinder is inextensible and in unsteady motion. Its axis may be slightly curved, with radius of curvature large in comparison with the cylinder radius.Two model problems are considered in detail to investigate the effect of lateral displacements of a cylinder with an established boundary layer. The boundary layer on the flexible cylinder is found to be shed once the lateral displacement of the cylinder axis exceeds the boundary-layer thickness. The unsteady pressures generated by this vortex shedding are investigated.

The second variation of area of minimal surfaces in four-manifolds

1993 ◽  
Vol 295 (1) ◽  
pp. 245-267 ◽  
Author(s):  
Mario J. Micallef ◽  
Jon G. Wolfson
Keyword(s):  
Minimal Surfaces ◽  
Second Variation ◽  
Four Manifolds ◽  
The Second Variation

Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials

2004 ◽  
Vol 69 (6) ◽  
Author(s):  
D. Bria ◽  
B. Djafari-Rouhani ◽  
A. Akjouj ◽  
L. Dobrzynski ◽  
J. P. Vigneron ◽  
...  
Keyword(s):  
Band Structure ◽  
Band Gap ◽  
Photonic Band Gap ◽  
Lamellar Structures ◽  
Left Handed ◽  
Omnidirectional Photonic Band Gap ◽  
Photonic Band

SEM/FESEM imaging of lamellar structures in melt extruded polyethylene films

1992 ◽  
Vol 50 (2) ◽  
pp. 1142-1143
Author(s):  
R.T. Chen ◽  
M.G. Jamieson ◽  
R. Callahan
Keyword(s):  
Imaging Techniques ◽  
Membrane Surface ◽  
High Stress ◽  
Extrusion Direction ◽  
Polymeric Membranes ◽  
Scanning Tunneling ◽  
Tunneling Microscopy ◽  
Crystalline Polymers ◽  
Lamellar Structures ◽  
Resolution Imaging

“Row lamellar” structures have previously been observed when highly crystalline polymers are melt-extruded and recrystallized under high stress. With annealing to perfect the stacked lamellar superstructure and subsequent stretching in the machine (extrusion) direction, slit-like micropores form between the stacked lamellae. This process has been adopted to produce polymeric membranes on a commercial scale with controlled microporous structures. In order to produce the desired pore morphology, row lamellar structures must be established in the membrane precursors, i.e., as-extruded and annealed polymer films or hollow fibers. Due to the lack of pronounced surface topography, the lamellar structures have typically been investigated by replica-TEM, an indirect and time consuming procedure. Recently, with the availability of high resolution imaging techniques such as scanning tunneling microscopy (STM) and field emission scanning electron microscopy (FESEM), the microporous structures on the membrane surface as well as lamellar structures in the precursors can be directly examined.The materials investigated are Celgard® polyethylene (PE) flat sheet membranes and their film precursors, both as-extruded and annealed, made at different extrusion rates (E.R.).

Correlation averaging of two-dimensional crystals: basic strategy and refinements

1986 ◽  
Vol 44 ◽  
pp. 136-139
Author(s):  
W. Baumeister ◽  
R. Rachel ◽  
R. Guckenberger ◽  
R. Hegerl
Keyword(s):  
Low Dose ◽  
Signal To Noise Ratio ◽  
Real Space ◽  
Fourier Domain ◽  
Two Dimensional ◽  
Signal To Noise ◽  
Unit Cells ◽  
Lateral Displacements ◽  
Established Technique ◽  
Crystalline Array

IntroductionCorrelation averaging (CAV) is meanwhile an established technique in image processing of two-dimensional crystals /1,2/. The basic idea is to detect the real positions of unit cells in a crystalline array by means of correlation functions and to average them by real space superposition of the aligned motifs. The signal-to-noise ratio improves in proportion to the number of motifs included in the average. Unlike filtering in the Fourier domain, CAV corrects for lateral displacements of the unit cells; thus it avoids the loss of resolution entailed by these distortions in the conventional approach. Here we report on some variants of the method, aimed at retrieving a maximum of information from images with very low signal-to-noise ratios (low dose microscopy of unstained or lightly stained specimens) while keeping the procedure economical.

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